Question

Which of the following equations represent a pair of perpendicular straight lines?
A. ${y^2} + xy - {x^2} = 0$
B. ${y^2} - xy + {x^2} = 0$
C. ${x^2} + xy + {y^2} = 0$
D. ${x^2} + xy - 2{y^2} = 0$

Answer 1

Hint: A pair of straight lines, passing through the origin, are represented by a general equation of the form $a{x^2} + 2hxy + b{y^2} = 0$ . Sum of the slopes of the two lines is given by $\dfrac{{ - 2h}}{b}$ and the product of the slopes is given by $\dfrac{a}{b}$ . The angle between the two lines, $\theta $ , is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$. We will use this formula to derive the condition and use it to get the desired solution.

Formula Used: The angle between a pair of straight lines, passing through the origin, represented by $a{x^2} + 2hxy + b{y^2} = 0$ is calculated using the formula $\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ .

Complete step-by-step solution:
Let us consider a general pair of straight lines, passing through the origin.
$a{x^2} + 2hxy + b{y^2} = 0$ … (1)
Let the angle between the straight lines be $\theta $ .
Now, we know that the tangent of the angle between them is given by the formula:
$\tan \theta = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$ … (2)
For the lines to be perpendicular to each other, $\theta = \dfrac{\pi }{2}$ .
Substituting this in equation (2), we get:
$\tan \dfrac{\pi }{2} = \infty = \left| {\dfrac{{2\sqrt {{h^2} - ab} }}{{a + b}}} \right|$
From this we can conclude that:
$a + b = 0$
This gives:
$a = - b$ … (3)
Hence, for a pair of straight lines to be perpendicular to each other, $a = - b$ .
Now, we’ll consider each of the equations given in the provided options and check if they are perpendicular or not.
For option A:
${y^2} + xy - {x^2} = 0$
Comparing this equation with equation (1), we get:
$a = - 1,b = 1$
That means $a = - b$
From equation (3), we know that for two lines to be perpendicular $a = - b$ , which is satisfied with the equation given in option A.
While for equations given in other options, this condition was not satisfied.
Thus, the correct option is A.

Note: The above question can be solved even faster when the condition for the lines to be perpendicular is known. Thus, it is advised to a student to understand and learn the conditions required for a pair of lines to be perpendicular, parallel and coincident.